While thinking about various ways of deriving
scales from the harmonic series, I wondered "Why derive a scale,
why not just use the harmonic series itself?" This guitar is the
result of that thought.

Due to the limited range of the guitar, it was not
practical to begin with the lowest string tuned to the first
partial: doing so would result in the lower half of the
instrument's range consisting of just a few notes. By tuning the
lowest string to the 4th partial, then the other strings can be
tuned to partials 6,8,12,16, and 24, the same open-fifths tuning
used on the 21-tone Just Intonation Guitar, and this gives a
good spread and playable partials from 4 up to 48.

The color scheme is a little different from the
21-Tone Just Intonation guitar, since there are many more
prime-limit families. I tried to follow a basic spectrum
arrangement (the colors in the above
photo are not very accurate, it makes the greens and blues look
the same):

2 limit - red

3 limit - orange

5 limit - yellow

7 limit - green

11 & 13 limit - blue

17 limit - purple

19 and higher - violet

The Prime Guitar

The first guitar I refretted, (the 21-Tone Just
Intonation Guitar), used a 21 note scale derived from the
harmonic series, using pitches up to the 7 prime limit, and with
the scale repeating throughout the range of the instrument.
After that, I wanted a guitar that would enable me to play in
the harmonic series without limit-ations, so I next refretted an
instrument into the Harmonic Series Guitar. Being able to play
partials like 11, 13, 17 and other higher primes was quite
interesting, and I found myself wondering if an instrument could
be designed that would play only higher prime partials.

The Prime Guitar is designed to play prime
partials 17-199. The lowest string is tuned to the 17th partial
of a series, and the successive strings are then tuned to
partials 23, 37, 47, 67, and 89 to preserve the basic tonal
spread of a standard set of guitar strings. Since every single
note is a unique prime limit pitch, it was not practical to
color code the frets in the same way as the previous guitars. On
the other hand, it is very difficult to orient one's self on a
neck with such irregular fret spacing without any color coding,
so I settled on a simple alternating three color scheme that
allows for easy visual spotting of where the frets are located.

Because the lowest partial is 17, there are no
familiar lower prime limit intervals like octaves (2/1), fifths
(3/2), thirds (5/4), etc. Nevertheless, some intervals between
higher primes approximate the lower prime limit intervals (e.g.
61/31 is very close to an octave), without of course reproducing
them exactly. This gives the guitar a unique sound that can be
heard in this piece composed and recorded on this instrument:

(dedicated to Dorota Bartniczuk, who
generously donated the guitar that was converted into the
Prime Guitar)

The Diatonic Harmonic Guitar

The next idea was to explore a limited part of the basic
harmonic series in more detail. I chose to first look at the 4th
octave, or 8-15. This gives an 8-note scale that I wanted repeated
in each octave of the guitar so I could see what dyads and chords
were possible with just this small scale. I say "small" because 8
notes is only one more than the diatonic collection that is used
in traditional western music, and is therefore not unlike the
white keys on the piano in terms of number of notes and scale
degrees. For this reason I thought a good name for this octave
would be the "Diatonic Harmonic" scale. Similarly, the next octave
(16-31) has sixteen notes, similar to (in some ways, and also so
different from, in others) our familiar chromatic scale, so it
could be called the "Chromatic Harmonic" scale. The octaves above
these, with a never ending, increasing plethora of notes, await
other names.

This scale has some notes that are very familiar: besides the
tonic (1/1), there is a major second (9/8), a major third (5/4), a
perfect 5th (3/2), and the harmonic minor 7th (7/4) which, while
quite different in tuning from the equal tempered minor 7th, is
still recognizable as a minor seventh. There is also the quite
normal sounding major 7th (15/8). Of course, all these notes are
tuned in just intonation, so they differ from their familiar
tempered versions by very little (the major second and perfect
fifth) or somewhat more (the major third and minor seventh).

Then there are two rather unfamiliar notes: 11 and 13. These are
two primes and therefore introduce new families of notes which
will reproduce higher in the harmonic series. Here, however, they
make their first appearance. 11 (or 11/8) is an almost exact
quarter-tone between what we call a perfect fourth and a tritone
in equal temperament. As such, it sounds a bit like both, and a
bit like neither. It is the first truely alien sounding interval
we find in the harmonic series, since 7, while technically outside
our musical system as it historically developed, does not jar the
ear to the same extent that 11 and 13 do. 13 is a kind of "neutral
sixth" although it is somewhat closer to the equal tempered minor
sixth, it still sounds very bizarre, especially in combination
with the more familiar notes. To me this combination of 6
relatively familiar notes with two alien ones is what makes this
octave of the harmonic series quite interesting sounding.

Another thing to consider is the richness of intervallic variety
between the notes of this scale. While there are only 8 notes,
these produce no less that 48 unique intervals in their various
combinations. Recall that the 12 notes of our equal temperament
system can only produce 11 unique intervals (assuming octave
equivilence of course). It is this great variety, I believe, that
makes it possible to get quite a lot of interesting music out of
just these 8 notes. One can only be staggered at the thought of
the variety of sounds and colors that are available in the next
octave with its 16 notes, and higher still. The possibilites for
musical expression await any musician willing to open their ears,
listen, and then create. There is much work to be done, and we are
just at the tip of the iceburg. Rather than feeling the creative
exhaustion of post-20th century 12 tone equal temperament, we can
look to an unlimited future of new sounds that should keep us busy
for many generations to come.